On the proof of the solvability of a linear problem arising in magnetohydrodynamics with the method of integral equations

The paper is concerned with a linear system of Fredholm–Volterra singular integral equations arising in the study of a linearized initial-boundary value problem of magnetohydrodymnamics for a fluid surrounded by an infinite vacuum region. It is proved that this system is solvable in the class of continuous functions satisfying the Hölder condition with respect to the spatial variables, which yields a classical solution of the problem in question. §1. Statement of the problem and main result Let Ω1 be a bounded simply connected domain with boundary S of class C , α ∈ (0, 1), and let Ω2 = R 3 \ s Ω1. In Ω1 ∪Ω2, we consider the following initial-boundary value problem for the vector field H(x, t), x ∈ Ω1 ∪ Ω2: (1.1) ⎧⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ μ1Ht(x, t) + α −1 curl curlH(x, t) = 0, divH(x, t) = 0, x ∈ Ω1, t ∈ (0, T ), curlH(x, t) = 0, divH(x, t) = 0, x ∈ Ω2, H τ −H τ = a(x, t), μ1H · n− μ2H · n = b(x, t), x ∈ S, H → 0, |x| → ∞, H(x, 0) = 0, x ∈ Ω1 ∪ Ω2. Here μ1, μ2, α are positive constants, n is the exterior unit normal to S with respect to Ω1, H (i) = H|x∈s Ωi , H (i) τ = H (i) − n(H · n) is the tangential component of H, i = 1, 2, a, b are given functions on S × (0, T ) = ST , a · n = 0. For simplicity, we set α = μ−1 1 , so that the first equation in (1.1) becomes Ht + curl curlH = 0. We assume that a(x, 0) = 0, b(x, 0) = 0. Problems like (1.1) arise in the analysis of the problems of magnetohydrodynamics where the magnetic and electric fields should be found not only in the domain Ω1 filled with fluid but also in the surrounding vacuum region Ω2 (see [1, 2, 3]). In the present paper we obtain the classical solution of (1.1) by reducing this problem to a system of singular integral equations on S of mixed type. In the case where Ω2 = ∅, such a 2010 Mathematics Subject Classification. Primary 45B05.